Solve for $X$. $\left[\begin{array}{rr}4 & -21 \\ 13 & 6 \\8 &-9\end{array}\right]+X=\left[\begin{array}{rr}7 & -16 \\ 12 & 19 \\1 &0\end{array}\right] $ $X=$
Solution: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}4 & -21 \\ 13 & 6 \\8 &-9\end{array}\right]+X=\left[\begin{array}{rr}7 & -16 \\ 12 & 19 \\1 &0\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}4 & -21 \\ 13 & 6 \\8 &-9\end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}7 & -16 \\ 12 & 19 \\1 &0\end{array}\right] $ Then we can rewrite the equation as follows. $A+X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A+X&=B\\\\ X&=B-A\end{aligned}$ Finding $X$ We found that $X=B-A$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=B-A \\\\&=\left[\begin{array}{rr}7 & -16 \\ 12 & 19 \\1 &0\end{array}\right] -\left[\begin{array}{rr}4 & -21 \\ 13 & 6 \\8 &-9\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(7-4) & (-16+21) \\ (12-13) & (19-6) \\(1-8) &(0+9) \end{array}\right] \\\\\\&=\left[\begin{array}{rr}3 & 5 \\ -1 & 13 \\-7 &9\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}3 & 5 \\ -1 & 13 \\-7 &9\end{array}\right]$